In the theory of von Neumann algebras, a subfactor of a factor. The theory of subfactors led to the discovery of the Jones polynomial in knot theory.

Index of a subfactor

, so that it has a finite trace. In this case every Hilbert space which is a non-negative real number. The index obtained from the GNS construction.

Jones index theorem

. All these values occur.

Basic construction

is called the basic construction.

.

Jones tower

factors of finite index. By iterating the basic construction we get a tower of inclusions

.

which satisfy the Temperley–Lieb relations. Whenever these extra conditions are satisfied, the algebra is called a TemperlyLieb.

Standard invariant

.

The standard invariant is the following grid:

which is a complete invariant in the amenable case. A diagrammatic axiomatization of the standard invariant is given by the notion of planar algebra.

Principal graphs

is said to be irreducible if either of the following equivalent conditions is satisfied:

• bimodule;

• the relative commutant.

. The relative tensor product, described in and often called Connes fusion after a prior definition for general von Neumann algebras of Alain Connes by decomposing the following tensor products into irreducible components:

bimodules arising in this way form the vertices of the principal graph, a bipartite graph on the right. The dual principal bimodules.

Since any bimodule corresponds to the commuting actions of two factors, each factor is contained in the commutant of the other and therefore defines a subfactor. When the bimodule is irreducible, its dimension is defined to be the square root of the index of this subfactor. The dimension is extended additively to direct sums of irreducible bimodules. It is multiplicative with respect to Connes fusion.

The subfactor is said to have finite depth are hyperfinite, Sorin Popa is isomorphic to the model

factors are obtained from the GNS construction with respect to the canonical trace.

Knot polynomials

with the relations above is called the Temperley–Lieb algebra. This is a quotient of the group algebra of the braid group, so representations of the Temperley–Lieb algebra give representations of the braid group, which in turn often give invariants for knots.

References

Bibliography

• Theory of Operator Algebras III by M. Takesaki